1. Introduction
High-category tropical cyclones (in categories higher than 3) are known as typhoons in the northwestern Pacific and South China Sea (SCS) and as hurricanes in the northeastern Pacific and North Atlantic. Such cyclones cause some of the worst natural disasters, and surface waves generated by typhoons [typhoon waves (TWs)] cause great economic losses and numerous human casualties every year. TWs destroy marine platforms, vessels, and ocean shorelines (Wang and Oey 2008). It is therefore important to model and forecast TWs in any attempt to minimize damage. On the other hand, TWs affect atmospheric models of tropical cyclones because meteorological models use information on surface waves to indicate surface roughness in order to improve the parameterization of air–sea momentum and heat fluxes (Liu et al. 2011; Wu et al. 2015, 2016). A coupled wind–wave–current model could improve the accuracy with which the intensities of tropical cyclones can be predicted. The study of TWs is therefore of considerable importance when developing a typhoon–ocean coupled system.
The first TW observations were made in the 1930s, when a composite chart of TW swell directions was produced (Tannehill 1936). However, early shipboard observations are treated as having low accuracy and being inadequate for current purposes (King and Shemdin 1978). Later in the 1970s, buoys were deployed to make observations of TWs by the U.S. National Oceanic and Atmospheric Administration (NOAA) National Data Buoy Center. These buoys have been collecting continuous in situ TW data since they were deployed, and the data allow TW research to be performed. In situ wave observations made on oceanic platforms also capture temporal variations in TWs (Forristall et al. 1978).
The study of TWs has also benefited from advances in aircraft-based and satellite-based remote sensing techniques. Remote sensing scans covering relatively large regions affected by TWs can be made from aircraft (King and Shemdin 1978) using synthetic aperture radar. Beal et al. (1986) analyzed satellite-based “Seasat” ocean wave spectra for a band 900 km wide, and they identified the probable areas in which TWs were generated. Later, Wright et al. (2001) and Walsh et al. (2002) studied sea surface directional wave spectra in all quadrants of the center of Hurricane Bonnie (1998) over open water and at landfall. The data were acquired by the U.S. National Aeronautics and Space Administration (NASA) Scanning Radar Altimeter (SRA) data. They found that dominant waves were generally not propagated downwind of the local wind and that the directional wave field in the vicinity of a hurricane may be inherently predictable and modeled using few parameters. They also developed a simple model to predict the dominant wave propagation direction.
Groundbreaking work on TW dynamics was published in the 1990s. Young (1998) investigated 229 one-dimensional frequency spectra for TWs for 16 hurricanes. Young (1998) found that TWs had a uniform one-peak frequency spectrum if the buoy station was less than 8Rmax (Rmax is the radius of maximum sustained wind) from the eye of the cyclone. The spectra were consistent with fetch-limited wave spectra, which are realistic wave spectra generated through uniform and sustained wind forcing (Donelan et al. 1985). The term “fetch” refers to the effective forcing distance over which waves are generated.
Young (2006) later analyzed buoy-based, two-dimensional, frequency–directional wave spectra and again found one-dimensional wave spectra. Young (2006) also found that the two-dimensional wave spectra, which he also found, could be parameterized. Young (2006) determined shape parameters for JONSWAP spectra from TW observations in the Southern Hemisphere. The results of this study meant that an idealized two-dimensional frequency–directional TW wave spectrum could be constructed if the peak wave period (PWP) was known. However, it is still argued that directly parameterizing TWs does not allow TW mechanics to be represented adequately. For example, although the equivalent fetch (or PWP) of TW can be parameterized (Young 1988; Young and Vinoth 2013), their assumptions on the idealized wind and also the spatial distribution of TW limited their prediction skill on the level of low order. Meanwhile, the parameterization of wind-wave-dominated conditions near the center of a typhoon could not be used to give the entire wave field including the swell-dominated region. Further research is therefore required before TWs can be directly parameterized.
Dynamic wave models are widely used, unlike the direct parameterization method, to supply TW information. Third-generation wave models are currently the most state-of-the-art dynamic wave models. Third-generation wave models solve a frequency–directional wave spectrum by parameterizing the source terms that affect the energy and shape of the spectrum. The source terms wind input, nonlinear wave–wave interactions, and wave energy dissipation through whitecapping wave breaking are assumed to be important to TWs in the open ocean. The first third-generation wave model to be devised was called the Wave Model [WAM; Wave Model Development and Implementation (WAMDI); WAMDI Group 1988]. WAVEWATCH III (WW3; Tolman and Chalikov 1996; Tolman 2014) is a modification of WAM in which the effect of surface current on wave frequency is considered. The Simulating Waves Nearshore (SWAN; Booij et al. 1999) improves WAM through take account in the depth-forced wave breaking in the nearshore.
Third-generation wave models are widely used to hindcast and forecast waves. Huang et al. (2013) used the SWAN model to simulate TWs in the Gulf of Mexico, and they examined the effects of different wind input parameterizations on the modeling of TWs. Moon et al. (2003) used WW3 to simulate snapshot frequency–directional TW spectra. The significant wave height (SWH) in the open ocean simulated in the well-organized experiments of Moon et al. (2003) agreed well with aircraft-based and buoy-based observations, showing that the model had an encouraging ability to simulate wave heights. Third-generation wave models have also contributed to studies of TW dynamics. Liu et al. (2007) found that resonance between local winds and swells caused the storm translation speed to influence the asymmetric structures of TWs. Fan et al. (2009) studied surface current forcing using the WW3 model and found that the surface current can influence wave propagation and significantly affect the wave spectrum.
The performances of third-generation wave models in describing wave periods and wave spectra are, however, not acceptable. Xu et al. (2007) simulated TWs using the SWAN model nested within the WW3 model and found that the simulated 2D spectra failed to reproduce the narrow bands of swell waves. Compared with wind input and nonlinear wave–wave interactions, the whitecapping breaking dissipation term is little understood (Cavaleri et al. 2007). It is clear that there are still important deficiencies in wave models, and uncertainties in the source terms and wind forcing are still the main problems when modeling TWs. There is still some scope for a marked improvement in the performance of TW models, especially in terms of wave period (Xu et al. 2007). Some assessment of the effects of different parameterization schemes on TW models is also required (Kalantzi et al. 2009).
The effects of nonlinear wave–wave interactions on models of TW dynamics have still not been quantified very well, but these effects are assumed to have important effects on the shapes of TW spectra. Young (2006) proposed that a unimodal spectral shape is controlled by nonlinear interactions, giving a smooth transition between wind–wave interactions and swell and not generating separate bimodal peaks. Therefore, variations in the PWP or the shape of the wave spectrum have been assumed to be controlled by nonlinear wave–wave interactions even though such interactions do not contribute to the wave energy or SWH. The contributions of nonlinear interactions to TWs are not easy to determine by analyzing data, but they can be determined using numerical methods.
Continual in situ buoy-based wave observations are available for the eastern Pacific and the Gulf of Mexico, but observations in the SCS are reported only occasionally. We recently performed a study in the SCS aimed at improving our ability to forecast typhoons and TWs. Validating a wave model using in situ observations is the most fundamental step when building a regional atmosphere–wave–current coupled model. The work presented here was performed as part of the development of a regional wave model for the SCS. The aim of the study was to provide information on the underlying dynamics of TWs by combining observations and model outputs. The study was supposed distinctive for its well rebuilt wind forcing. We performed sensitivity tests on the wave model source terms and to determine the best performance of the typical third-generation wave model. We compared the results using different sets of source terms, and this revealed underlying variations in the TWs. The effects of nonlinear wave–wave interactions on TWs were assessed in isolation. The remainder of this paper is organized as follows: The three typhoons that were studied and the positions of the buoys in the array are described and the surface wave characteristics are analyzed in section 2. The model configuration and experimental design are described in section 3. The experiment results are presented in section 3. The effects of nonlinear wave–wave interactions on TWs are discussed in section 4, and our conclusions are drawn in section 5.
2. TW observations
a. Descriptions of the buoys and typhoons
We installed wave buoys in the SCS. Some buoys were damaged or destroyed in a severe storm, so three buoys (B0, B1, and B4, marked on Fig. 1) were used after quality control procedures had been performed. Each disk buoy (3 m in diameter) recorded wind vector and wave parameters at defined time intervals. A wind monitor (model 05013; R. M. Young Company, Traverse City, Michigan, United States) on each buoy measured the wind speed and wind direction, and a TRIAXYS wave sensor (AXYS Technologies, Sidney, Canada) on each buoy measured the SWH, PWP, and mean wave direction. The wind vector was measured 4 m (3 m) above the mean sea surface at B1 and B4 (B0) and was transformed to the standard height (10 m) using a bulk formula (Large and Pond 1981). It is noted that the relative error of wind speed transformation was limited to 10% if we considered the concept of saturation in the drag coefficient parameterizations (Powell et al. 2003; Hwang 2011; results are not shown here). The longitudes and latitudes of the three buoys used are given in Table 1. The water was more than 1500 m deep at the site of each of the three buoys. A depth of 1500 m is generally used to represent deep water when surface gravity waves are of interest. The three buoys captured data for three typhoons, which were Typhoons Hagupit (2008), Rammasun (2014), and Kalmaegi (2014). All three typhoons formed in the western Pacific.
Map of the study area and (inset) an expanded map of the South China Sea. Tracks of the three typhoons are shown as colored curves. The positions of the buoys are marked with circles. The black, blue, and red rectangles represent the three model domain layers.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
The longitudes and latitudes, water depths, and IPs of the buoys.
According to the Joint Typhoon Warning Center (JTWC) best-track records, Typhoon Hagupit (2008) formed on 17 September and ended on 25 September. Typhoon Hagupit formed farther north than the other typhoons studied but then moved southward, allowing it to come into contact with warmer sea surfaces and to gain more heat energy from the ocean than it would otherwise have done. Typhoon Hagupit entered the SCS through the Luzon Strait on 22 or 23 September and arrived at buoy B0 on 23 September. The minimum distance between buoy B0 and Typhoon Hagupit was about 55 km (see more information in Table 2).
Typhoon information around the buoys. The nondimensional time t is defined as the time relative to the typhoon passing by and then normalized by the local inertial period. The maximum wind speed, radius of maximum wind, and translation speed are denoted by Vmax, Rmax, and UH, respectively. The typhoon information is linearly interpolated from JTWC best-track data.
Typhoon Rammasun (2014) formed at 8.5°N and 152.9°E on 9 July. Typhoon Rammasun formed much farther east and south than the other two typhoons. Typhoon Rammasun entered the SCS by passing over the Philippine Islands. Less energy was supplied to the typhoon over the islands than would have been supplied over the ocean, so the islands decreased the intensity of the typhoon to some extent. The buoys were to the right of the track of Typhoon Rammasun, and the typhoon was closest to buoy B1 on 17 July. The best-track JTWC data showed that the minimum distance between buoy B1 and Typhoon Rammasun was 298 km.
Typhoon Kalmaegi (2014) formed 2 months after Typhoon Rammasun and much farther west than Typhoons Hagupit and Rammasun. Similar to Typhoon Rammasun, the eye of Typhoon Kalmaegi crossed the Philippine Islands and then entered the SCS. Typhoon Kalmaegi first made landfall in the Philippine Islands on 14 September. Typhoon Kalmaegi was still very intense as it entered the SCS after crossing the Philippine Islands. Crossing the Philippine Islands did, however, decrease the intensity of Typhoon Kalmaegi somewhat. Typhoon Kalmaegi passed through the center of the buoy array and was recorded by buoys B1 and B4. Both buoys were to the right of the track of Typhoon Kalmaegi.
b. Analysis of the TW observations
In light of the results of previous studies, we transformed the longitude–latitude coordinates into a reference grid that moved with the storm. We made simultaneous wind and wave observations, which allowed more reliable analyses to be performed than was possible in previous studies using only wave observations (Young 2006). We restricted our assessment to the forcing stage of each typhoon, defined as half the inertial period (IP) before and after the time the typhoon arrived (the time the typhoon was closest to the buoys). We therefore assumed that the TW signals before and after the time period covering the forcing stage of a typhoon were not related to the typhoon.
The wind and wave direction distributions are shown in Fig. 2a. The wind directions in Typhoon Kalmaegi in the right-front quadrant were generally consistent with the results of the idealized wind vortex model in that the wind direction was across the track and leftward when the buoy was in front of the typhoon track and along the track and forward when the typhoon arrived. This means that the wind direction rotated clockwise as the typhoon approached the buoy. However, the wave direction was more forward than the wind direction before the typhoon arrived, and the angle between the two directions was about 20°. The wave direction rotated in the same sense as the wind direction, but more slowly, when the typhoon approached and was then almost the same as the wind direction when the typhoon arrived. There was an abnormal point near the edge of the 10Rmax forcing area, where the wind direction was rightward and forward. The wind direction at this point was clearly at odds with the results of the idealized wind vortex model. This instantaneous abnormal wind direction was probably related to the incorrect eye location from best-track data. It is noted the best-track data generally smooth over deviations in track and speed and frequently miss sharp jumps in eye location in the real world (Wright et al. 2009).
(a) Vectors indicating the mean wave directions for the three sets of buoy data during the three typhoons, with dotted vectors showing the local wind directions. (b) Locations dominated by wind sea (nondimensional peak frequency ν > 0.13; Fig. 3) are indicated by circles, and locations dominated by swell (ν < 0.13) are indicated by solid dots. The data are projected onto a frame of reference moving with the storm. The storm center is at the center of each figure. The coordinates represent the distance to the storm center scaled by the radius of the maximum wind speed Rmax. The propagation direction of each typhoon is toward the top of the page.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
The Typhoon Rammasun data points were also projected to the right-front quadrant even though the data were collected far from the eye of the typhoon. The wind direction was generally forward, but the wave direction was more leftward. The difference between the wind and wave directions was clear when the buoy was in front of or to the right of the eye of the typhoon. The wind direction was more likely to be consistent with the wave direction when the eye of the typhoon was to the left of the buoy.
The buoys only interrogated the left side of Typhoon Hagupit. The wind direction in the left-front quadrant was generally backward and leftward outside the 5Rmax area and only rightward within the 5Rmax area. The mean wave direction was leftward outside the 5Rmax area and backward within the 5Rmax area. The wind direction when the eye of the typhoon was closest to the buoy could not be explained by a symmetrical idealized wind vortex, indicating that there was some uncertainty in the wind structure of the typhoon. The Typhoon Hagupit data were also projected into the left-rear quadrant, in which the wind direction was only rightward when the typhoon arrived and then rotated counterclockwise as the typhoon passed over the buoy. In contrast, the wave directions remained backward and rightward as the typhoon passed over the buoy.
The wind and wave directions in the right-rear quadrant are very plausible during Typhoon Kalmaegi. The wind direction was forward when the eye of the typhoon was close to the buoy and then rotated clockwise as the typhoon moved away. The wave direction was slightly different from the wind direction, but it aligned with the wind direction in the right-rear quadrant near 10Rmax. The wave direction during Typhoon Rammasun in the right-rear quadrant was generally along the track of the typhoon and was clearly at an angle to the wind direction.
We compared our observations with the results of Young (2006) even though we used parameters with slightly different definitions. We used the mean wave direction, whereas Young (2006) used the dominant wave direction. We would argue that the two directions are comparable because the wave direction distribution in a two-dimensional wave spectrum is dominated by one main direction (Fig. 2 in Young 2006). Some of the features we found were consistent with those found by Young (2006). For example, the mean wave direction was closely aligned with the local wind direction in the right-rear quadrant. However, Young (2006) noted that waves in the left-front and left-rear quadrants are likely to radiate out from a region to the right of the eye of the typhoon, but no clear signs of this were found in our data. This could have been because our data for the left-front and left-rear quadrants were only for Typhoon Hagupit, which passed much closer to the buoy than was the case in the study performed by Young (2006). Additionally, using the mean wave direction probably damped some of the signals found in the dominant wave direction data used by Young (2006).
We investigated our observational data using fetch-limited theory, as proposed by Young (2006). It is first worth mentioning, however, that typhoon forcing and fetch-limited forcing are different. Wind sea and swell are distinguished mainly by the wave age in fetch-limited forcing because the wind-sea and swell directions are aligned. Wind forcing rotates and is transient, and there is an angle between the wind-sea and swell directions under typhoon conditions. Wind forcing may not have any noticeable effects on TW energy as the wind sea of TWs grows. Investigating TW wave ages is therefore necessary. Donelan et al. (1985) proposed that an inverse wave age U10/Cp of 0.83 should be seen as the limit defining the transition between wind sea and swell. In that equation, U10 is the wind speed at 10 m, and Cp is the phase speed of the waves at the spectral peak frequency.
The spatial distributions of the wind sea and swells in all quadrants are shown in Fig. 2b. It can be seen that the wind and wave directions differ in the forcing areas of the typhoons (within 5Rmax of the typhoon centers) but that the wave ages almost remained within the growing wind-sea range defined in fetch-limited theory. The wave ages in the forward parts of the typhoon tracks and outside the forcing areas of the typhoons (beyond 5Rmax of the typhoon centers) showed that the TWs were almost dominated by the relatively mature swells based on fetch-limited theory. Waves in the left-rear quadrant were dominated by remote swell generated farther than 10Rmax from the typhoon center. Waves in the right-rear quadrant were almost always generated locally. Our results differed slightly from those of a previous study, in which it was suggested that the spectra are dominated by swell in all quadrants of a hurricane except for the right-rear quadrant (Young 2006).
We validated Eq. (1) using our observational data, and the results are shown in Fig. 3. The vertical line at ν = 0.13 in Fig. 3 is equivalent to U10/Cp = 0.83. The growing wind sea is assumed to occur at ν > 0.13. It can be seen from Fig. 3 that our observations are in remarkably good agreement with the empirical relationship when the TWs were dominated by the wind-wave component (ν > 0.13). However, there were marked differences between our observations and the empirical relationship at ν < 0.13, the observations obviously lower than the empirical values. The data differ from a fetch-limited sea by up to a factor of 10. The main disagreements between our observations and the empirical relationships related to the data for the front of Typhoon Hagupit (as shown in Fig. 2b). Therefore, the fetch-limited theory may overestimate the energies of TWs at ν < 0.13, especially in the left-front parts of the typhoon track and outside the forcing area of the typhoon (10Rmax).
Nondimensional energy as a function of the nondimensional peak frequency. Data observed during the three different typhoons are represented using different symbols. The oblique line is the relationship proposed by Donelan et al. (1985). The vertical line at ν = 0.13 is the commonly used demarcation between swell and wind sea.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
3. TW modeling
a. Model configuration
We modeled TWs using WW3 (version 4.18), which is a state-of-the-art third-generation wave model. The WW3 model describes the wave state using the wave action density spectrum and uses source terms to determine temporal variations in the wave action density spectrum. Source terms represent forcing and dissipation in the wave system, and there is an energy redistribution term that refers to nonlinear wave–wave interactions. In the open ocean, the total source term S has three main components: a wind–wave interaction term Sin, a nonlinear wave–wave interaction term Snl, and term Sds describing dissipation through whitecap breaking (Tolman 2014).
We used multiple-nested domains in the wave model (Fig. 1). The scope of the first region was the northwest Pacific Ocean (the area within the black rectangle in Fig. 1), and the horizontal grid resolution was 0.25°. The swell signal introduced to the SCS from the northwest Pacific Ocean was of primary concern in the first region. The second domain level (region; with a resolution of 0.125°) was nested within the first region. The second domain covered the SCS (the area within the blue rectangle in Fig. 1). This region was a transitional region used to avoid an excessive resolution gap between the first and third regions and to take into account the topography of the SCS. The third region (with a resolution of ⅞°), within the second domain, was the area from 15° to 23°N and from 110° to 120°E (the area within the red rectangle in Fig. 1). This region covered the domain we were most concerned with and included the tracks of the typhoons of interest and the positions of the buoys. The directional resolution in each domain was 10°, and the frequency range was 0.04–0.4 Hz with an increment factor of 1.1. The spinup time for wave modeling was 1 month.
It is possible for blended wind data to give incorrect wind data for the site of a buoy, and this could make it impossible to validate the model. This could have occurred because the shape parameter B, an exponential decay parameter, was not correct. The ambient pressure pn, which is strongly related to B, was not given by the best-track data, so pn had to be set empirically. We attempted to identify a reasonable value of pn using the buoy observations to minimize uncertainty in the values of pn and B. We used the bisection method to identify the roots of the equations. Another possible reason for the inconsistency of the wind data was our use of imperfect best-track data. For example, as a special case, the observed wind during Typhoon Hagupit had a double peak, implying that the eye of the typhoon passed over the buoy. The JTWC data showed that the minimum cross-track distance was approximately Rmax. Paradoxically, the observed wind speed was dramatically lower than Vm. This meant that the JTWC data could have underestimated the real Rmax or overestimated Vm. The rebuilt wind data for a specific location were found to be very sensitive to the values of Rmax and Vm. Therefore, Rmax and Vm were slightly adjusted (to 1.3Rmax and 0.8Vm, respectively) for Typhoon Hagupit around the buoy during the forcing time (−0.5 to 0.5 IP relative to the typhoon arrival time). The rebuilt and observed wind speeds are shown in Fig. 4. It can be seen that the rebuilt wind speeds were quite consistent with the observed wind speeds. This is because the ambient pressure was adjusted adaptively and because the Rmax and Vm values for the special case for Typhoon Hagupit were treated with caution.
The rebuilt and observed wind speeds at the buoy stations. The x axis is time t relative to the typhoon passing by, that is, t = (T − t0)/IP, where T is the real time, t0 is the time the typhoon passed closest to the buoy, and IP is the inertial period.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
As mentioned earlier, we performed sensitivity tests on the source term schemes. Three input and dissipation source term schemes were examined in the present study, namely, the Tolman and Chalikov (1996) scheme (TC96; Tolman and Chalikov 1996), the WAM 4+ and Ardhuin et al. (2010) scheme (WAM4P&A10), and the Babanin–Young–Donelan–Rogers–Zieger scheme (BYDRZ; Donelan et al. 2006; Young and Babanin 2006; Babanin et al. 2007, 2010; Rogers et al. 2012; Zieger et al. 2015). TC96 built the wave breaking–induced dissipation term by the sum of the low-frequency part and high-frequency part. WAM4P&A10 then developed more physical-sounding dissipation terms by combination of saturation-based spontaneous breaking and the short-wave dissipation induced by longer breaker. Besides, WAM4P&A10 (also later BYDRZ) introduced explicit swell dissipation to activate the swell dissipation under the no wind condition. BYDRZ parameterized more reliable source terms based on the observation in a lake. Their input source terms considered three new mechanics, which included the airflow separation that depressed the wind input, the nonlinear dependence of the energy growth rate on the wave spectrum, and also the enhancement of energy input by wave breaking. In the idealized duration-limited experiment, WAM4P&A10 released more dissipation around the peak frequency as compared with TC96 for the condition of growing sea state, while BYDRZ delivered significantly amplified high-frequency dissipation, which was used to balance the stronger energy input by the new scheme. BYDRZ (TC96) showed the strongest (weakest) wave energy in the idealized experiment (Zieger et al. 2015). As for the TWs, the case study of Hurricane Katrina revealed that BYDRZ gave the highest wave energy (or SWH), while TC96 led to the smallest wave period (Zieger et al. 2015).
b. Model results
1) SWH time series
The SWH time series is shown in Fig. 5. The results of the numerical model agreed well with the observations in general. The simulated SWHs for Typhoon Hagupit were slightly underestimated before the arrival of the typhoon. As the typhoon built to its maximum intensity, the SWHs simulated using the BYDRZ scheme were a little higher than observed, but the SWHs simulated using the TC96 scheme were lower than observed. After the typhoon had passed the buoy, the SWHs simulated using the BYDRZ scheme remained higher than observed but the SWHs simulated using the WAM4P&A10 and TC96 schemes came closer to the observations. The best performing parameters during the forcing stage of Typhoon Hagupit (−0.5 to 0.5 IP, relative to the typhoon arrival time) gave a mean relative bias of 18.8% and a root-mean-square error (RMSE) of 1.462 m. The best performance was found when the TC96 scheme was used (Table 3).
Simulated and observed significant wave heights at the buoy stations. The x axis is time t relative to the typhoon passing by.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
Statistics for the relationships between the simulated and observed significant wave height data.
The model results and observations were somewhat different for the buoy on the outlying right-hand side of the track of Typhoon Rammasun. The three source schemes all gave markedly different results during the peak forcing stage of the typhoon. The maximum modeled SWH was >6.7 m using all three source schemes, but the maximum observed SWH was about 5 m. The simulated SWHs agreed well with the observations before and after the typhoon, the differences between the simulated and observed SWHs being <0.5 m. The best performing scheme (WAM4P&A10) during the forcing stage gave a mean relative bias of 42%.
The model results were better for Typhoon Kalmaegi than for the other typhoons. The buoys were to the right of the typhoon track in the intensive forcing area. The simulations underestimated the SWHs throughout the simulation period, but the differences between the simulated and observed SWHs were negligible. The best performing scheme (BYDRZ) gave mean relative biases of −5.1% at B1 and −0.7% at B4 and RMSEs of 0.998 m at B1 and 0.76 m at B4 during the forcing stage. There were, however, marked differences between the maximum SWHs. The errors at the two buoys used were around 2 m at the time the strongest forcing occurred, and the errors were higher before the strongest forcing and negative afterward.
In general, WW3 prefers to underestimate the right-side wind wave slightly, as in the case of Kalmaegi. However, if the buoy is located on the right side but relatively far from the typhoon center (case Rammasun), the TW is dominated by swell, and then the WW3 shows a risk of overestimating the SWH significantly, even though the magnitude of wind forcing is consistent with the observation. There is no unanimous best scheme for three typhoons. The results suggest the specific scheme should be different for each case, and the choice is probably related to the relative location of buoy to typhoon track.
2) PWP time series
The simulated PWPs for the three typhoons are shown in Fig. 6 and Table 4. The observed PWP in the forcing stage of Typhoon Hagupit was about 14 s, but the highest simulated PWP was 17 s. The frequency determined using the model was lower than observed, so the simulated waves had longer wavelengths than the actual waves. The PWP simulations before and after the typhoon agreed well with the observations. The PWPs determined using the different source term schemes were not clearly different but could be distinguished. At a specific time, but not at the peak forcing time, the maximum difference between the PWPs was about 3 s. The mean error for the average PWP was <0.2 s for all three source term schemes. The best performing scheme (TC96) gave a mean relative bias of 7.8%.
As Fig. 5, but for the peak wave period.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
Statistics for the relationships between the simulated and observed peak wave period data.
The buoy closest to Typhoon Rammasun was relatively far from the core of the typhoon, but the PWP still had a single peak, which showed that no significant swell occurred before the typhoon arrived. The modeled time series matched the observations, but the magnitudes of the model results and observations were different. The simulated PWPs were higher than the real PWPs. The TC96 scheme performed the best and gave a mean relative bias of 22.6% and a RMSE of 2.61 s.
The PWP measured by buoy B4, to the right of the Typhoon Kalmaegi track and not far from the core of the typhoon, increased sharply just before the typhoon arrived. The background PWP was 8 s, and the PWP increased to 13 s for 1 day. Low-frequency waves are a strong signal of swell generated in a typhoon forcing region and propagating to great distances in front of the typhoon when it is still far from a buoy (Wang and Oey 2008). Such a sharp increase cannot easily be replicated in a numerical model, and no sharp increase was found using any of the three source term schemes. However, no clear sharp increase was found at the other buoy (B1) taking measurements of Typhoon Kalmaegi, so the simulation for buoy B1 was relatively good for the stage before the typhoon arrived. The PWP simulation during the forcing stage was generally consistent with the observations, although some results were lower than the actual PWP values. Using the TC96 scheme, the PWP RMSE in the forcing stage was 1.79 s for buoy B1 and 1.867 s for buoy B4. The relative errors using the different source term schemes were 0.3–0.75 s, and the best performing scheme was BYDRZ.
The BYDRZ scheme gave the largest PWPs in all four time series, the WAM4P&A10 scheme gave moderate PWPs, and the TC96 scheme gave the smallest PWPs. We conclude that, to achieve this, the BYDRZ scheme helps WW3 to generate a relative low-frequency wave, or equivalently a longer wavelength wave, than other two schemes. BYDRZ are therefore supposed to release special source terms, regardless of the relative buoy locations to the typhoon track.
3) Temporal variations in wave spectra
We now investigate the underlying TW mechanics that control temporal variations in wave spectra, that is, the source terms. Temporal variations in the one-dimensional wave spectra and the three important source terms (the wind–wave interaction term Sin, the dissipation term Sds, and the nonlinear interaction term Snl, and the sum of all three) are shown in Fig. 7 (for Typhoon Hagupit) and Fig. 8 (for Typhoon Kalmaegi).
Variations in the one-dimensional wave spectra, the wind–wave interaction term Sin, the dissipation term Sds, the nonlinear interaction term Snl, and the sum of the three terms over time at buoy location B0 during Typhoon Hagupit. A term to be tested is in each row, and each column is one parameterization scheme. The thick black lines represent peak frequencies. The x axis is the frequency, and the y axis is time t relative to the typhoon passing by.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
As Fig. 7, but for buoy B1 during Typhoon Kalmaegi.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
The spectra and source terms for Typhoon Hagupit, for which the buoy was to the left of the track and within the Rmax of the eye of the typhoon, are shown in Fig. 7. The energy spectrum for scheme TC96 had the narrowest frequency coordinate distribution. The wave energy concentrates around the peak frequency, which leads to similar spectrum shape among three schemes. The surface wave was dominated by swell with a frequency of about 0.05 Hz during the first half of the forcing period, and the swell energy increased as the typhoon approached. Wind waves became dominant once the typhoon arrived at the station, and the peak spectrum moved sharply to a high frequency. The high-frequency wave was ephemeral. Later, in the second half of the forcing period, the wave energy decayed and the peak wave frequency shifted leftward. There were two peak frequencies, showing that double-peak wind forcing was induced by the eye of the typhoon. The swell in the first half of the forcing stage contained more energy and the frequency bandwidth of the swell was broader when the WAM4P&A10 scheme was used than when the TC96 scheme was used. Information on the eye of the typhoon was not shown in the variations in peak frequency. The energy distribution contained a broader frequency band when the BYDRZ scheme was used than when the other two schemes were used, not only for the swell before the typhoon arrived but also for the wind waves generated by typhoon forcing.
The Sin term could clearly explain wind-wave generation. The Sin value was trivial during the first half of the forcing stage, but the Sin value changed markedly when the typhoon was very close to the station. The Sin value affected the higher-frequency band more than the peak frequency when wind waves dominated. The Sin values produced by the three different schemes were clearly different. The BYDRZ scheme supplied more energy at high frequencies than the other schemes, and this caused the frequency bandwidth in the spectrum structure to be broader when the BYDRZ scheme was used than when the other schemes were used.
In terms of Sds, all three schemes decreased wave energy at frequencies near to and higher than the peak frequency. The Sds determined using the TC96 scheme suggested that stronger dissipation occurred at higher frequencies, whereas the Sds determined using the other two schemes suggested that dissipation occurred mainly around the peak frequency.
Very different nonlinear Snl terms were found when the different schemes were used even though the nonlinear term schemes were the same. The nonlinear term depended on the wave spectrum, so different wave spectra produced using different input and dissipation source terms caused different nonlinear terms to be found. The same Snl formula was used when each scheme was used, so the nonlinear terms showed the same pattern (wave energy shifted from higher frequencies to lower frequencies) when each scheme was used. The differences were in the magnitudes of the Snl terms. The TC96 scheme gave the weakest nonlinear transformation, but the BYDRZ scheme gave the strongest nonlinear transformation.
Energy gains and losses were finally determined by the sums of the three terms. The wind-wave spectrum was strongly related to the source term balance. In the wind-wave-dominated situation, the three schemes gave markedly different source terms. The BYDRZ scheme gave the most energy input around the peak frequency and at higher frequencies. The BYDRZ scheme still showed energy inputs occurring at higher frequencies when the peak frequency reached its maximum value. However, energy losses at those frequencies occurred when the other two schemes were used. More energy inputs occurring when the BYDRZ scheme was used than when the other schemes were used led to the SWH being 1 m higher during the wave energy decaying period (Fig. 5a).
The spectra and source terms for Typhoon Kalmaegi, at buoy B1, are shown in Fig. 8. The buoy was to the right of the typhoon track, 3.9Rmax from the typhoon center. The most notable feature was the initial peak frequency, which was in the high-frequency band, implying that significant nonlocal swell was absent before the typhoon arrived. The peak frequency moved downward and the wavelength increased when the wind intensified. The wave spectrum reached maximum energy at around the typhoon arrival time. The wave energy decreased after that, and the peak wave frequency increased. The wave spectrum was different when each different source term scheme was used. The maximum wave spectrum value was smaller when the BYDRZ scheme was used than when the TC96 and WAM4P&A10 schemes were used.
Temporal variations in the Sin and Sds values were consistent when the three different schemes were used. Energy gains and losses mainly occurred very close to the typhoon arrival time. However, high-frequency waves contributed more, and energy input was always stronger at a high frequency (0.15 Hz) than at the peak frequency, when the BYDRZ scheme was used than when the other schemes were used. Energy inputs were concentrated in the vicinity of the peak frequency when the other two schemes were used. The Sds when the BYDRZ scheme was used dissipated at high frequencies, and energy losses were not as important at the peak frequency as at high frequencies. The Sds when the TC96 and WAM4P&A10 schemes were used dissipated more at the peak frequency than at higher frequencies, although dissipation at high frequencies was important.
Differences between the wave spectra determined using the three schemes therefore induced clear differences in the Snl. The Snl value had different magnitudes when different schemes were used. The Snl was smaller when the TC96 scheme was used than when the other schemes were used. The Snl term helped waves gain energy at frequencies slightly higher than the peak frequency just before the typhoon arrival time (0.05 IP relative to the typhoon arrival time) when the TC96 scheme was used. The Snl term generally led to energy loss at frequencies higher than the peak frequency when the other schemes were used. This was strongly related to the energy being stored at high frequencies when the BYDRZ scheme was used.
The sum of the three source terms explained the evolution of the wave spectrum, how wave energy was gained around the peak frequency, and how energy was lost at high frequencies. Less energy was gained around the peak frequency when the BYDRZ scheme was used than when the other two schemes were used. The positive value frequency bandwidth was broader when the BYDRZ scheme was used than when the other two schemes were used, implying that more frequencies were involved in the TW simulation when the BYDRZ scheme was used than when the other two schemes were used.
4. Discussion
a. Effect of nonlinear wave–wave interactions on the TW model
We designed a twin experiment to quantify the roles of nonlinear wave–wave interactions on TW dynamics. The control run used Typhoon Hagupit and the TC96 scheme. The test run is the same except that the nonlinear wave–wave interaction terms have been set to zero. The spectra and source terms of the experiments are shown in Fig. 9a. The results of the control run are therefore the same as that in Fig. 7. In terms of temporal variations in the wave spectrum, the most important point was that the test run wave spectrum was more energetic than the control run at high frequencies (0.15–0.3 Hz). The peak frequency in the test run was clearly higher than the peak frequency in the control run. The peak frequency of the test run suddenly decreased to a low frequency at around the typhoon arrival time, indicating that swell then became dominant. The input and dissipation source terms used in the test run mainly affected the higher frequencies. The sum of the input and dissipation source terms therefore determined the gains and losses in the wave spectrum. The energy dissipated to a remarkable degree between 0 and 0.1 IP. However, nonlinear interactions helped the control run have a remarkable positive energy input, given the sum of the Sin and Sds effects around the peak frequency and at 0.1 IP. The SWH was higher in the test run than in the control run and was a considerable overestimate in view of the observed SWH (Fig. 9b). The maximum SWH in the test run was 13.54 m, which was 2.53 m higher than the observed SWH. Nonlinear interactions did not cause a net energy change in the wave spectrum. The wave spectrum without nonlinear interactions was more efficient at absorbing energy from the wind.
(a) Variations in the one-dimensional wave spectra, the wind–wave interaction term Sin, the dissipation term Sds, and the sum of the terms over time for buoy location B0 during Typhoon Hagupit. A term to be tested is in each column. The control run is in the first row, and the test run in the second row. (b) Simulated significant wave heights for the twin experiments and observations at buoy station B0. The x axis is the relative time t. (c) One-dimensional wave spectra for the experiments at the typhoon arrival time.
Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0174.1
Most importantly, nonlinear interactions helped cause a solitary peak in the one-dimensional wave spectrum at the typhoon arrival time. The test run without nonlinear interactions generated a double-peak structure for the wave spectrum (Fig. 9c). Without a sustained transfer of energy from high to low frequencies, the wave spectrum in the test run could easily be decomposed into two components: a relatively high-frequency wave near 0.2 Hz and a relatively low-frequency wave near 0.08 Hz. This bimodal wave spectrum was therefore a pseudobimodal spectrum, but it indicates the important role of nonlinear interactions in the formation and evolution of TWs.
b. Sensitivity of horizontal resolution on model results
We examined the sensitivity of the model performance on model grid resolution for the three-layer nested grid. The results supported the simulated SWH and PWP time series not being modified by the use of different grid resolutions (data not shown). The horizontal resolution was thus not critical to the TW simulation, if the information of wind forcing was not enriched from the coarse grid to the fine grid. However, in some coupled model simulations where the wind forcing has a delicate high-resolution grid, the fine grid in the wave model is thus needed to maintain the necessary spatial variation.
5. Conclusions
A series of buoy observations of TWs in the SCS were used to make some observations in support of the following conclusions: First, the mean wave directions were closely aligned with the local wind directions in the right-rear quadrant. However, in most other parts of the study area, there were clearly differences between the mean wave directions and local wind directions. These results were consistent with the results of previous studies. Second, in the right-front quadrant of the typhoon center, the mean wave direction was affected by the distance between the buoy and the typhoon core. At the buoy for which the minimum distance to the typhoon core was <5Rmax during Typhoon Kalmaegi, the mean wave direction rotated clockwise from the local wind direction by about 20°. At the buoy that was relatively far from the center of Typhoon Rammasun, the mean wave direction probably rotated counterclockwise to the wind direction. Third, we distinguished between wind sea and swell using the inverse wave age U10/Cp = 0.83 as a limit to define the transition between wind sea and swell. We found that most of the waves in the front part of the typhoon track within 5Rmax were dominated by locally generated wind waves. Waves were dominated by swell beyond 5Rmax. In the left-rear quadrant at >10Rmax, the waves were dominated by swell. The waves in the right-rear quadrant were almost always dominated by wind waves even at >10Rmax. Fourth, we reconfirmed that for a growing wind sea, the relationship between the total energy of the spectrum and the peak frequency in the TW data agreed remarkably well with the relationship for fetch-limited wind waves, in other words a growing wind sea could be explained by fetch-limited theory. However, the disagreements between fetch-limited theory and the observational results are sometimes large, and the main points of disagreement were projected to the front part of Typhoon Hagupit, which was dominated by swell. Therefore, for swell, the fetch-limited theory may overestimate the energies of TWs, especially in the left-front parts along the typhoon track and outside the main forcing area.
We designed a set of experiments using the WW3 model to investigate the ability of the model to describe TWs and the influences of the source terms on the modeled TWs. Three typhoons were considered, and the buoys were in different locations in each case relative to the tracks of the different typhoons. To better simulate TWs, we rebuilt near-realistic wind fields by correcting the idealized wind vortex with in situ observations. The model results were compared with the buoy observations. The numerical results agreed well with the observations, except for the case where the buoy was located on the right side but relatively far from the typhoon center (case Rammasun). Taking the source terms into consideration, our results revealed that the best model gave a mean relative bias of −0.7% and a RMSE of 0.76 m for the SWH and a mean relative bias of −3.4% and a RMSE of 1.115 s for the PWP.
There was some uncertainty in the parameterization of the source terms used in the wave model. Our numerical experiments indicated that the source terms used affected the wave parameters. The SWH and PWP simulations were affected by the wind–wave interaction term and the dissipation term. Three schemes, the TC96 scheme, the WAM4P&A10 scheme, and the BYDRZ scheme, were considered. The different schemes gave markedly different source terms in the wind-wave-dominated case. The BYDRZ scheme gave the strongest energy input around the peak frequency and at higher frequencies. The large energy gain found using the BYDRZ scheme was responsible for that scheme giving a higher SWH than the other schemes in the time series analysis. The nonlinear interaction scheme remained the same in all the experiments, but the nonlinear interaction source terms were also changed because the wave spectra were different, due to the effects of the three types of source terms on the wind input and whitecapping dissipation.
We also designed twin numerical experiments to investigate the effects of nonlinear wave–wave interactions on the TW model. Nonlinear wave–wave interactions affected the structure of the wave spectrum. Nonlinear wave–wave interactions efficiently transferred wave energy from high to low frequencies and stopped a double-peak structure from forming in the frequency-based spectrum. The missing nonlinear interaction source term led to more energy gain occurring in the wave spectrum and a higher SWH than in the control run with nonlinear interactions.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (41476021 and 41576013), the National Basic Research Program of China (2013CB430302), and the Indo-Pacific Ocean Environment Variation and Air–Sea Interaction project (GASI-IPOVAI-04). We are grateful to Weifang Jin and Chujin Liang for postprocessing the observational data. We also thank the two anonymous reviewers for their constructive comments.
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